Binomial expression an expression (a + b)n which is the sum of two terms raised to the power n.
e.g. (x + 3)2
Binomial expansion (a + b)n expanded into a sum of terms
e.g. x2 + 6x + 9
Binomial expansions get increasingly complex as the power increases:
![Binomial expansion](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-8.25.25-PM.png)
The general formula for each term in the expansion is nCr an-r br .
In order to find the full binomial expansion of a binomial, you have to determine the coefficient nCr and the powers for each term. The powers for an and b are found as n − r and r respectively, as shown by the binomial expansion formula.
Binomial expansion formula
DB 1.9
![Binomial expansion formula](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.03.32-PM-1024x173.png)
The powers decrease by 1 for a and increase by 1 for b for each subsequent term.
The sum of the powers of each term will always = n.
There are two ways to find the coefficients: with Pascal’s triangle or the binomial coefficient function (nCr). You are expected to know both methods.
Pascal’s triangle
![Pascal’s triangle 1](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.08.19-PM.png)
Pascal’s triangle is an easy way to find all the coefficients for your binomial expansion. It is particularly useful in cases where:
- the power is not too high (because you have to write it out manually)
- you need to find all the terms in a binomial expansion
Binomial coefficient functions
The alternative is to calculate the individual coefficients using the nCr function on your calculator, or with the formula below.
(Note:In the 1st term of the expansion r = 0, in the 2nd term r = 1, . . .)
![Binomial coefficient functions](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.13.22-PM.png)
Expanding binomial expressions
![Expanding binomial expressions](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.19.03-PM.png)
Use the binomial expansion formula
![Expanding binomial expressions 1](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.33.41-PM-1024x325.png)
2. Find coefficients using Pascal’s triangle for low powers or nCr on calculator for high powers
![Expanding binomial expressions 2](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.37.44-PM.png)
3. Put the terms and their coefficients together
![Expanding binomial expressions 3](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.42.41-PM-1024x122.png)
4. Simplify using laws of exponents
![Expanding binomial expressions 4](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-07-at-9.48.53-PM.png)
Finding a specific term in a binomial expansion
Find the coefficient of x5 in the expansion of (2x − 5)8
Use the binomial expansion formula
2. Determine r
Since a = 2x, to find x5 we need a5.
a5 = an−r = a8−r, so r = 3
3. Plug r into the general formula
4. Replace a and b
8C3 (2x)5 (−5)3
5. Use nCr to calculate the value of the coefficient, nCr
![Finding a specific term in a binomial expansion](https://revisiontown.com/wp-content/uploads/2023/03/IMG_20230308_131503-1024x516.jpg)
6. Substitute and simplify
56 × 25(x5) × (−5)3 = −224000(x5)
⇒ coefficient of x5 is −224000
The IB use three different terms for these types of question which will effect the answer you should give.
Coefficient the number before the x value
Term the number and the x value
Constant term the number for which there is no x value (x0)